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Prove that quadratic equation (bx+c)^2-{(a^2)(x^4)}=0 has at least 2 real roots for all a,b,c belonging to real numbers.

Prove that quadratic equation (bx+c)^2-{(a^2)(x^4)}=0 has at least 2 real roots for all a,b,c belonging to real numbers.

Grade:11

1 Answers

Ritesh Khatri
76 Points
3 years ago
Let f(x) = (bx+c)^2-{(a^2)(x^4)}
f(0) = (b0+c)^2-{(a^2)(0^4)} = c2 
now we know f(0) is positive 
f(-c/b) = (b(-c/b)+c)^2-{(a^2)((-c/b)^4)} = – {(a^2)((-c/b)^4)}
So  f(-c/b) is negative. 
since f is polynomial it is always continous and it changes sogn from + to  – hence by IVT it must have root between -c/b and 0.

also degree of f id 4 and nonn real roo occur in pairs hence 1 real root imply another real root
 

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